Respuesta :

josephtainilai2

Solution :

[tex]x = \frac{4 + \sqrt{43} }{2} .x = 4 + \frac{4 - \sqrt{43} }{2} [/tex]

Step-by-step explanation:

[tex] \frac{1}{(x - 5)} + \frac{3}{(x - 2)} - 4[/tex]

1. Multiply by LCM

[tex]x = 2 + 3( x - 5) = 4 - (x - 5)(x + 2)[/tex]

2. Solve

[tex]x = 2 + 3(x - 5) = 4 (x - 5)(x + 2)[/tex]

[tex]x = \frac{4 + \sqrt{43} }{2} .x = 4 + \frac{4 - \sqrt{43} }{2} [/tex]

3.Verify Solutions

Find undefined (singularity) points :

x=5,x=–2

[tex]x = \frac{4 + \sqrt{43} }{2} .x = 4 + \frac{4 - \sqrt{43} }{2} [/tex]

[tex]\\ \rm\Rrightarrow \dfrac{1}{x-5}+\dfrac{3}{x+2}=4[/tex]

[tex]\\ \rm\Rrightarrow \dfrac{x+2+3x-15}{(x-5)(x+2)}=4[/tex]

[tex]\\ \rm\Rrightarrow 4x-13=4(x-5)(x+2)[/tex]

[tex]\\ \rm\Rrightarrow 4x-13=4(x(x+2)-5(x+2))[/tex]

[tex]\\ \rm\Rrightarrow 4x-13=4(x^2+2x-5x-10)[/tex]

[tex]\\ \rm\Rrightarrow 4x-13=4(x^2-3x-10)[/tex]

[tex]\\ \rm\Rrightarrow 4x-13=4x^2-12x-40[/tex]

[tex]\\ \rm\Rrightarrow 4x^2-16x-53=0[/tex]

On solving we get

[tex]\\ \rm\Rrightarrow x=2\pm\dfrac{69}{2}[/tex]

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